Decoherence of cold atomic gases in magnetic microtraps

Decoherence of cold atomic gases in magnetic microtraps

C. Schroll, W. Belzig, and C. Bruder

Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland 共Received 13 May 2003; published 15 October 2003兲

We derive a model to describe decoherence of atomic clouds in atom-chip traps taking the excited states of the trapping potential into account. We use this model to investigate decoherence for a single trapping well and for a pair of trapping wells that form the two arms of an atom interferometer. Including the discrete spectrum of the trapping potential gives rise to a decoherence mechanism with a decoherence rate⌫ that scales like

⌫⬃1/r₀⁴with the distance r0from the trap minimum to the wire.

DOI: 10.1103/PhysRevA.68.043618 PACS number共s兲: 03.75.Dg, 03.75.Gg, 03.65.Yz I. INTRODUCTION

Cold atomic gases form an ideal system to test fundamen- tal quantum-mechanical predictions. Progress in laser cool- ing made it possible to achieve previously inaccessible low temperatures in the nK range共see, e.g., Ref.关1兴兲. One of the most exciting consequences of this development has been the creation of Bose-Einstein condensates 共BECs兲 关2,3兴. These have been manipulated by means of laser traps in various manners, e.g., vortices have been created or collisions of two BECs have been studied 关4 – 6兴. An interesting link to solid- state phenomena has been established by creating optical lat- tices, in which a Mott transition has been theoretically pre- dicted关7兴and observed关8兴.

Recently, proposals to trap cold atomic gases using micro- fabricated structures 关9兴 have been realized experimentally 关10–12兴 on silicon substrates, so-called atom chips. These systems combine the quantum-mechanical testing ground of quantum gases with the great versatility in trapping geom- etries offered by the microfabrication process. Microfabri- cated traps made it possible to split clouds of cold atomic gases in a beam-splitter geometry 关13兴, to transport wave packets along a conveyer belt structure关14兴, and to accumu- late atomic clouds in a storage ring 关15兴. Moreover, a BEC has been successfully transferred into a microtrap and trans- ported along a waveguide created by a current-carrying mi- crostructure fabricated onto a chip 关16 –19兴. Finally, several suggestions to integrate an atom interferometer for cold gases onto an atom chip have been put forward关20–22兴.

The high magnetic-field gradients in atom-chip traps pro- vide a strongly confined motion of the atomic quantum gases along the microstructured wires. The atomic cloud will be situated in close vicinity of the chip surface. As a conse- quence, there will be interactions between the substrate and the trapped atomic cloud, and the cold gas can no longer be considered to be an isolated system. Recent experiments re- ported a fragmentation of cold atomic clouds or BECs in a wire waveguide 关19,23–25兴 on reducing the distance be- tween the wave packets and the chip surface, showing that atomic gases in wire traps are very sensitive to its environ- ment. Experimental关26兴and theoretical works 关27兴showed that there are losses of trapped atoms due to spin flips in- duced by the strong thermal gradient between the atom chip, held at room temperature, and the cold atomic cloud.

In principle, both fluctuating environments and atom-

atom interactions may lead to decoherence of the atomic mo- tion. In Refs. 关28,29兴 the influence of magnetic near fields and current noise on an atomic wave packet were studied.

Consequences for the spatial decoherence of the atomic wave packet were discussed under the assumption that the trans- versal states of the one-dimensional waveguide are frozen out.

For wires of small width and height, as used for atom- chip traps, the current fluctuations will be directed along the wire. Consequently, the magnetic-field fluctuations generated by these current fluctuations are perpendicular to the wire direction, inducing transitions between different transverse trapping states.

In this paper we will discuss the influence of current fluc- tuations in microstructured conductors used in atom-chip traps. We derive an equation for the atomic density matrix that describes the decoherence and equilibration effects in atomic clouds taking transitions between different transversal trap states into account. We study decoherence of an atomic state in a single waveguide as well as in a system of two parallel waveguides. Decoherence effects in a pair of one- dimensional 共1D兲 waveguides are of particularly high inter- est as this setup forms the basic building block for an on-chip atom interferometer关20–22兴.

Our main results can be summarized as follows. Consid- ering current fluctuations along the wire, we show that spa- tial decoherence along the guiding axis of the 1D waveguide occurs only by processes including transitions between dif- ferent transversal states. The decoherence rate obtained scales with the wire-to-trap distance r₀as⌫⬃1/r₀⁴. Applying our model to a double waveguide shows that correlations among the magnetic-field fluctuations in the left and right arms of the double waveguide are of minor importance. The change of the decoherence rate under the variation of the distance between the wires is dominated by the geometric rearrangement of the trap minima.

To arrive at these conclusions we proceed as follows. Sec- tion II will give a derivation of the kinetic equation describ- ing the time evolution of an atomic wave packet in an array of N parallel 1D waveguides subject to fluctuations in the trapping potential. We will then use this equation of motion in Sec. III to study the specific cases of a single-1D wave- guide and a pair of 1D waveguides. Spatial decoherence and equilibration effects in the single and double waveguide will

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be discussed. Finally, we summarize and give our conclu- sions in Sec. IV.

II. EQUATION FOR THE DENSITY MATRIX The trapping potential of a microstructured chip is pro- duced by the superposition of a homogeneous magnetic field and the magnetic field induced by the current in the conduc- tors of the microstructure. Atoms in a low-field seeking hy- perfine state 兩S典 will be trapped in the magnetic-field mini- mum关30兴. The magnetic moment of the atom is coupled to the magnetic field through the Zeeman interaction:

V共x兲⫽⫺具^S兩␮兩S典^B共x兲. 共1兲 As we consider traps with characteristic frequencies much lower than the Larmor precession frequency of the trapped atom, we assume that atoms prepared in a pure hyperfine state兩m_F典 will follow the magnetic field adiabatically. Hence we replaced the magnetic-moment operator ␮ ^{in Eq.} 共1兲 by its mean value具^S兩␮兩S典^.

Many different wire-field configurations will lead to an atom trap 关31,32兴. We will concentrate on systems in which the trapping field is produced by an array of parallel wires, see Fig. 1 for a single- or double-wire trap. A homogeneous bias field is applied parallel to the surface on which the wires are mounted. An example for the resulting field distribution for the single-wire trap is shown in Fig. 2.

A. Stochastic equation for the density matrix

The quantum-mechanical evolution of a cold atomic cloud is described by a density matrix. The time evolution of the density matrix ␳^(x,x

⬘

,t) is given by the von Neumann equation

iប ⳵

⳵^t␳⫽关H,␳兴, 共2兲

where H is the Hamiltonian of an atom in the trapping po- tential:

H⫽ p²

2m⫹V_t共r_⬜兲⫹␦^V共x,t兲. 共3兲 Here, the confining potential is assumed to be slowly varying along the longitudinal direction. Hence, it will be approxi- mated by V_t(r_⬜) which is taken to be locally constant along zˆ. The coordinate r_⬜denotes the coordinate perpendicular to the direction of the current-carrying wire or the waveguide.

The last term␦V(x,t) is a random fluctuation term induced by the current noise.

We will formulate the problem for a system of N parallel quasi-1D magnetic traps generated by a set of M parallel wires on the chip 共in general, N⫽M since some of the magnetic-field minima may merge兲. The number of parallel trapping wells is included in the structure of the confining potential V_t(r_⬜). In all further calculations we will assume that the surface of the atom chip is in the xˆ-zˆ plane and that the atoms are trapped in the half space y⬎0 above the chip.

The wires on the atom chip needed to form the trapping field are assumed to be aligned in zˆ direction.

In position representation for the density matrix, the von Neumann equation reads

iប ⳵

⳵^t␳共x,x

⬘

^,t兲⫽

冋

^⫺2mប2

冉

^dxd22⫺dxd2

^⬘

²

冊

^{⫹Vt共r⬜兲⫺Vt共r⬜}

^⬘

^兲

⫹␦^V共x,t兲⫺␦^V共x

⬘

^,t兲

册

^␳共x,x

^⬘

^{,t兲. 共4兲}

To derive a quasi-1D expression for Eq. 共4兲 we expand the density matrix in eigenmodes of the transverse potential

␹n(r_⬜):

FIG. 1. Setup of the atom chip showing directions of the wires and the magnetic bias fields needed to form the trapping potential.

共a兲Single-wire trap.共b兲Double-wire trap.

FIG. 2. Contour plot of the magnetic field of the single-wire trap for B_bias^(x)⫽10 G, B_bias^(z)⫽5 G, and I⫽0.1 A. The upper共right兲 plot shows a horizontal 共vertical兲 cut through the potential minimum.

The dashed lines in the upper and the right plot show the harmonic approximation to the trapping field.

␳共x,x

⬘

^,t兲⫽

兺

_n,m ^{␹n*共r⬜兲␹m共r⬜}

^⬘

^{兲␳nm共z,z}

^⬘

^{,t兲. 共5兲}

Here, the channel index n labels the transverse states of the trapping potential. The transverse wave functions␹n(r_⬜) are chosen mutually orthogonal and are eigenfunctions of the transverse part of the Hamiltonian in Eq.共3兲in the sense that

冋

^{⫺2mប2 ⵜr2⬜⫹Vt共r⬜兲}

册

^{␹n共r⬜兲⫽En␹n共r⬜兲. 共6兲}

Decomposition 共5兲 in transverse and longitudinal compo- nents of the density matrix is now inserted in the von Neu- mann equation, Eq.共4兲. Making use of the orthogonality and the completeness of the transverse states␹n(r_⬜) we obtain a one-dimensional equation for the evolution of the density matrix:

冋

^{iប⳵⳵t⫹2mប2}

冉

^dzd22⫺dzd2

^⬘

²

冊

^⫺⌬Elk

册

^␳lk共z,z

^⬘

^,t兲

⫽

兺

_n ^{关␦Sln共z,t兲␳nk共z,z}

^⬘

^{,t兲⫺␦Snk共z}

^⬘

^{,t兲␳ln共z,z}

^⬘

^,t兲兴.

共7兲 Here the abbreviation ⌬E_lk⫽E_l⫺E_k was introduced for the difference of transverse energy levels, and the fluctuations

␦^Snk(z) are defined as

␦^Snk共z,t兲⫽

冕

^{dr⬜␹n共r⬜兲␦V共x,t兲␹k*共r⬜兲. 共8兲}

The left-hand side of the reduced von Neumann equation, Eq.共7兲, describes the evolution of an atom wave function in the nonfluctuating trapping potential and we will hereafter abbreviate this part by (iប⳵t⫺Hˆ

lk). The terms on the right- hand side of Eq. 共7兲 contain the influence of the potential fluctuations.

The fluctuations of the confining potential are described by the matrix elements ␦^Snk(z,t) which imply transitions between different discrete transverse energy levels induced by the fluctuating potential. The influence of the fluctuating potential the longitudinal motion of the atomic cloud is in- cluded in the z dependence of␦^Snk(z,t).

B. Averaged equation of motion for the density matrix In this section we derive the equation of motion describ- ing the evolution of the density matrix具␳典averaged over the potential fluctuations 关33,34兴. To keep the expressions com- pact we rewrite Eq.共7兲as

共iប⳵t⫺H˜兲␳⫽␦^˜S␳^, 共9兲 where the spatial coordinates x, x

⬘

, time t, and all indices have been suppressed. The components of the quantities H˜ and␦˜ are defined as^S

H˜_{lki j}⫽Hˆ_lk␦li␦k j, 共10兲

␦^˜Slki j⫽␦^Sli共z,t兲␦k j⫺␦^Sjk共z

⬘

^,t兲␦li. 共11兲

The product in Eq. 共9兲 has the meaning A˜␳⫽兺i jA_{lki j}␳i j. The stochastic equation共9兲for the density matrix is averaged over the fluctuating potential using the standard cumulant expansion关34兴. Writing the time arguments again we obtain

共iប⳵t⫺H˜兲具␳共t兲典

⫽具␦^˜S共t兲典具␳共t兲典^⫺_បⁱ

冕

^0tdt

^⬘具具

^{␦˜S共t兲e⫺(i/ប)H˜ t⬘}

⫻␦^˜S共t⫺t

⬘

兲典典^{e(i/ប)H˜ t⬘}具␳共t兲典^. 共12兲 Here, the brackets具•典 denote the averaging over all realiza- tions of the potential fluctuations and the double brackets 具具•典典 denote the second cumulant. We can take the mean value of the potential fluctuation 具␦^V典 to zero, since the static potential is already included in the Hamiltonian on the left-hand side of Eq. 共12兲 and, hence, the first term on the right-hand side of Eq. 共12兲vanishes.

Reinserting the explicit expressions for H˜ ,␦^S˜ leads to the desired equation of motion for the density matrix

共iប⳵t⫺Hˆ_lk兲具␳lk共z,z

⬘

^,t兲典

⫽⫺ i

ប i jmn

兺 冕

0 t

d␶

冕

⫺⬁

⬁

dz˜

冕

⫺⬁

⬁

dz˜

⬘

^Ki j共z⫺˜,zz

⬘

⫺˜z

⬘

^,t⫺␶兲

⫻关具␦^Sli共z,t兲␦^Sim共˜,z ␶兲典␦k j␦jn

⫹具␦^Sjk共z

⬘

^,t兲␦^Sn j共˜z

⬘

^,␶兲典␦li␦im

⫺具␦^Sli共z,t兲␦^Sn j共˜z

⬘

^,␶兲典␦im␦jk

⫺具␦^Sjk共z

⬘

^,t兲␦^Sim共˜,z ␶兲典␦jn␦li兴具␳mn共˜,z˜z

⬘

^,␶兲典^. 共13兲 The kernel K_{i j}(z⫺˜,zz

⬘

⫺˜z

⬘

,t) is the Fourier transform of

K_{i j}共q,q

⬘

^,t兲⫽exp

冉

^{⫺i2mប 共q2⫺q}

^⬘

^{2兲t⫺ប ⌬i Ei jt}

冊

^{, 共14兲}

which can be explicitly evaluated, but this has no advantage for our further discussion.

Equation 共13兲 is the main result of this section. It de- scribes the evolution of the density matrix in a quasi-1D waveguide under the influence of an external noise source. It is valid for an arbitrary form of the transverse confining po- tential, thus, in particular, for single- and double-wire traps.

The main input is the external noise correlator and its effect on the trapped atoms. The noise correlator depends on the concrete wire configuration. Below we will derive a simpli- fied form of the equation of motion 共13兲under the assump- tion that the time scale characterizing the fluctuations is much shorter than the time scales of the atomic motion.

C. Noise correlation function

We will now derive the noise correlator for our specific system and use Eq. 共13兲 to study the dynamics. As we are interested in the coherence of atoms in an atom-chip trap, we will consider the current noise in the wires as the decoher- ence source. Fluctuations of the magnetic bias fields B_bias^(x) and B_bias^(z) , needed to form the trapping potential, will be ne- glected.

Using the approximation of a 1D wire, the fluctuating current density can be written as

j共x,t兲⫽I共z,t兲␦共x兲␦共y兲zˆ, 共15兲 where zˆ is the unit vector in zˆ direction. The fluctuations of the current density are defined by␦^j(x,t)⬅j(x,t)⫺具^j(x)典^{. It} is sufficient to know 具␦^I(z,t)␦^I(z

⬘

^,t

⬘

⁾典 to obtain the full current-density correlation function具␦^j(x)␦^j(x

⬘

⁾典. Note that the average currents are already included in the static poten- tial V_t(r_⬜).

The restriction of j to the zˆ direction is a reasonable as- sumption since we consider microstructured wires with small cross section A⫽l_wl_h, i.e., wires with widths l_wand heights l_h much smaller than the trap-to-wire distance r₀. Transver- sal current fluctuations lead to surface charging and thus to an electrical field which points in opposite direction to the current fluctuation. This surface charging effect will suppress fluctuations which are slow compared to␻RC⫽␴⌳/⑀0l_w re- sistance in transversal direction. Here, ␴ is the conductivity of the wire, ⑀0 the 共vacuum兲 dielectric constant, and ⌳

⬇1 Å is the screening length in the metal. This leads to RC frequencies of ␻RC⬇10¹³ Hz for wire widths of l_w

⫽10␮m and typical values for the conductivity in a metal.

The characteristic time scale for the atomic motion in the trap is given by the frequency of the trapping potential ␻

⬇10⁴ Hz. Thus, considering atomic traps with␻Ⰶ␻RC the current fluctuations can be taken along zˆ as a direct conse- quence of the quasi-one-dimensionality of the wire.

The current fluctuations are spatially uncorrelated as they have their origin in electron-scattering processes. Hence, the correlator具␦^I(z)␦^I(z

⬘

⁾典 has the form关35,36兴

具␦^I共z,t兲␦^I共z

⬘

^,t

⬘

兲典^⫽4kBTeff共z兲␴^A␦共z⫺z

⬘

兲␦c共t⫺t

⬘

兲. 共16兲 Here, k_Bis the Boltzmann constant. The effective noise tem- perature is given by 关35,36兴

T_eff共z兲⫽

冕

^{dE f共E,z兲关1⫺f共E,z兲兴, 共17兲}

where f (E,z) is the energy- and space-dependent nonequi- librium distribution function. A finite voltage across the wire induces a change in the velocity distribution of the electrons and the electrons are thus no longer in thermal equilibrium.

Nevertheless, the deviation from an equilibrium distribution is small at room temperature due to the large number of inelastic-scattering processes. The effective temperature T_eff accounts for possible nonequilibrium effects such as shot noise. However, contributions of nonequilibrium effects to

the noise strongly depend on the length L of the wire com- pared to the characteristic inelastic-scattering lengths. E.g., strong electron-phonon scattering leads to an energy ex- change between the lattice and the electrons. The nonequi- librium distribution is ‘‘cooled’’ to an equilibrium distribu- tion at the phonon temperature. Thus, nonequilibrium noise sources, such as shot noise, are strongly suppressed for wires much longer than the electron-phonon scattering length l_ep and the noise in the wire is essentially given by the equilib- rium Nyquist noise 关35,37,38兴. As the wire lengths used in present experiments are much longer than l_ep, we use T_eff

⬇300 K in all our calculations.

Finally,

␦c共t兲⫽1

␲

␶c

t²⫹␶c

2 共18兲

is a representation of the␦ function. The correlation time␶c

is given by the time scale of the electronic-scattering pro- cesses.

D. Simplified equation of motion

The dominating source of the current noise in the wires is due to the scattering of electrons with phonons, electrons, and impurities. These scattering events are correlated on a time scale much shorter than the characteristic time scales of the atomic system. This separation of time scales allows us to simplify the equation of motion 共13兲.

As a consequence of Eq.共16兲, the correlation function of the fluctuating potential will be of the form

具␦^V共x,t兲␦^V共x

⬘

^,t

⬘

兲典^⫽␦c共t⫺t

⬘

兲具␦^V共x兲␦^V共x

⬘

兲典^{. 共19兲} Using Eq. 共8兲we obtain

具␦^Sim共z,t兲␦^Sn j共z

⬘

^,t

⬘

兲典^⫽␦c共t⫺t

⬘

兲具␦^Sim共z兲␦^Sn j共z

⬘

兲典 共20兲 for the fluctuations of the projected potential.

We replace all the correlation functions in the equation of motion 共13兲by expression共20兲. The time integration can be performed using the fact that the averaged density matrix 具␳⁽␶⁾典 varies slowly on the correlation time scale␶c. Thus, 具␳⁽␶⁾典 can be evaluated at time t and taken out of the time integral. Finally, taking ␶cto zero, the Fourier transform of kernel 共14兲 leads to a product of ␦ functions in the spatial coordinates. Performing the remaining spatial integrations over z˜ and z˜

⬘

^{, Eq.}共13兲reduces to

共iប⳵t⫺Hˆ

lk兲具␳lk共z,z

⬘

^,t兲典

⫽⫺ i

2ប

兺

_mn ^{关具␦Sln共z兲␦Snm共z兲典具␳mk共z,z}

^⬘

^,t兲典

⫹具␦^Smk共z

⬘

兲␦^Snm共z

⬘

兲典具␳ln共z,z

⬘

^,t兲典

⫺2具␦^Slm共z兲␦^Snk共z

⬘

兲典具␳mn共z,z

⬘

^,t兲典兴. 共21兲 The trapping potential has N minima, i.e., the channel index n can be written as n⫽(␣^,nx,n_y), ␣⫽1, . . . ,N. We

will now assume that the minima are well separated in xˆyˆ direction such that we can neglect all matrix elements ␦^Si j

with different trap labels␣. Under this assumption, the first and second term on the right-hand side of Eq. 共21兲depend only on a single-trap label. In the third term, the trap labels may be different for␦^Slm as compared to ␦^Snk. Since cur- rent noise in one particular wire generates fluctuations of the magnetic field in all trapping wells, there will be a correla- tion between the fluctuations in different traps. It is hence this third term in Eq. 共21兲 which describes the correlation between potential fluctuations in different traps.

Using Eq.共21兲it is now possible to describe decoherence induced by current noise in trapping geometries of one, two, or more parallel wires关31,32兴. In the following sections, we will discuss decoherence in two specific trapping configura- tions: the single-wire trap in Sec. III A and the double-wire trap in Sec. III B, which is of particular interest for atom interferometry experiments关20–22兴.

III. TRAPPING GEOMETRIES

We will now consider two specific trap configurations, the single-wire trap and the double-wire trap. The dynamics of the noise-averaged density matrix will be discussed using Eq. 共21兲 which was derived in the preceding section. We assume that the wire generating the potential fluctuations is one dimensional. This assumption is reasonable for distances r₀of the trap minimum to the wire much larger than the wire width l_w and wire height l_h. Typical length scales are r₀

⬇ 10␮^{m–1 mm and wire widths and heights l}w

⬇10␮^m–50␮^{m, l}h⬇1 ␮^m关32兴.

To keep the notation short, we will suppress the brackets denoting the averaging. Thus, from now on, the density ma- trix␳i j denotes the average density matrix.

A. The single-wire trap

We are now looking at a specific magnetic trapping field having only a single one-dimensional trapping well as shown in Fig. 2. The magnetic field is generated by the superposi- tion of the magnetic field due to the current I in the single wire along the zˆ axis and a homogeneous bias field B_bias^(x) parallel to the chip surface and perpendicular to the wire共see Fig. 1兲. We additionally include a homogeneous bias field B_bias^(z) parallel to the trapping well. This longitudinal bias field is experimentally needed to avoid spin flips at the trap center.

Thus the magnetic field has the form

B共x兲⫽␮0I 2␲

1

x²⫹y²

冉

^⫺x0y

冊

^⫹

冉

^{BB0bias(x)bias(z)}

冊

^{. 共22兲}

The minimum of the trapping potential is located above the wire x₀⫽0 and has a wire-to-trap distance of r₀⫽y₀

⫽␮0I/(2␲^Bbias (x)).

We have now all the ingredients needed to calculate the fluctuation correlator 具␦^Sim(z)␦^Sn j(z

⬘

⁾典. Using Eqs. 共8兲, 共15兲, and 共16兲we obtain

具␦^Sim共z兲␦^Sn j共z

⬘

兲典⫽A_{imn j}J共z⫺z

⬘

兲. 共23兲 The transition matrix elements are given by

A_{imn j}⫽k_BT_eff␴^A

冉

^{␮0g␲F␮B BBbias(x)bias(z)}

冊

²

^冕

^{dr⬜␹i共r⬜兲共y⫺y0兲}

⫻␹m*_共r_⬜兲

冕

^dr⬜

^⬘

^␹n共r⬜

^⬘

^兲共y

^⬘

^{⫺y0兲␹j*共r⬜}

^⬘

^{兲. 共24兲}

The spatial dependence of the noise correlator is given by

J共z兲⫽ 1 r₀⁵

冕

⫺⬁

⬁

dz˜关1⫹˜z²兴^⫺3/2关1⫹共z/r₀⫺˜z兲²兴^⫺3/2. 共25兲 A more detailed derivation of the correlation function is given in the Appendix. The integration in Eq. 共24兲 can be done explicitly using harmonic-oscillator states for ␹n(r_⬜), which is a good approximation as long as B_bias^(z) is of the order of B_bias^(x) and the wire-to-trap distance r₀ is much larger than the transverse width w of the trapped state. Both require- ments are usually well satisfied in experiments. The dashed line in Fig. 2 shows the harmonic approximation to the trap- ping potential. Using the harmonic approximation we can characterize the steepness of the trapping potential by its trap frequencies. It turns out that the two frequencies coincide,

␻⫽

冑

^2mB␮Bbias^gF (z)

B_bias^(x)

y₀ , 共26兲

i.e., the trap potential can be approximated by an isotropic 2D harmonic oscillator 共2D HO兲.

After integration we obtain

A_{imn j}⫽A₀␦i_xm_x^˜␦i_ym_y␦n_xj_x^˜␦n_yj_y, 共27兲 where

␦

˜_i

ym_y⫽

冑

^my⫹1␦i_y,m_y⫹1⫹

冑

^my␦i_y,m_y⫺1, 共28兲 and the indices n_x, n_y denote the energy levels in xˆ and yˆ directions of the 2D HO. The prefactor A₀ is

A₀⫽k_BT_eff␴^A

冉

^{␮0g␲F␮B BBbias(x)bias(z)}

冊

^{2w22 共29兲}

and w⫽

冑

ប/(m␻) is the oscillator length of the harmonic potential.

Before moving on to derive the equation of motion for the averaged density matrix, let us discuss some consequences of expression 共27兲. Inspection of the coefficients A_{imn j} in Eq.

共27兲shows that there is no direct influence on the longitudi- nal motion of the atom, since the matrix element A_{ii j j} van- ishes. This result is not unexpected, because we consider only current fluctuations along the wire, which give rise to fluctuations in the trapping field along the transverse direc- tions of the trap potential. In fact, only transitions among energy levels of the yˆ component of the 2D HO give a

nonvanishing contribution. An explanation can be obtained by examining the change of the trap minimum position under variation of the current in the wire. Changing the current by ␦I leaves the trap minimum in xˆ direction unchanged at x₀⫽0, but shifts the yˆ -trap minimum by

␦^y0⫽␮0␦^I/(2␲^Bbias

(x)) 关39兴. Even though there is no direct coupling to the motion along the wire there can still be spa- tial decoherence in zˆ direction as we will see in the final result for ␳ obtained from Eq. 共21兲. However, this requires transitions to neighboring transverse energy levels.

Substituting Eqs. 共23兲 and 共27兲 in Eq. 共21兲 leads to an equation of motion for␳ for the single-wire trap configura- tion. Instead of discussing the general equation of motion of

␳ we will only consider the case where the two lowest- energy levels of the transverse motion are taken into account.

The restriction to the lowest-energy levels corresponds to the situation of the atomic cloud being mostly in the ground state of the trap and having negligible population of higher-energy levels. This situation is realistic, if the energy spacing of the discrete transverse states is large compared to the kinetic energy. Nevertheless, the result obtained from the two-level model provides a reasonable estimate for the decoherence of an atomic cloud, even if many transverse levels are popu- lated. Equation 共27兲 shows that only next-neighbor transi- tions are allowed. As A_{imn j}is a product of two next-neighbor transitions there can only be contributions of the next two neighboring energy levels. Higher-energy levels do only con- tribute to the decoherence by successive transitions which are of higher order in A_{imn j}and hence are negligible.

We rewrite Eq. 共21兲 for the subspace of the two lowest eigenstates in a matrix equation for the density vector

␳⫽共␳00,␳11,␳10,␳01兲. 共30兲 The indices of the averaged density matrix ␳lk denote the transverse state l⫽(l_x,l_y) but we are only writing the l_y component of the label as ␳ can only couple to states with the same x state 共i.e., only transitions between different y states of the 2D HO are allowed兲. The x label is l_x⫽0 for all states under consideration. Hence, Eq.共21兲can be written as a matrix equation for the density-matrix vector, Eq.共30兲,

关iប⳵t⫺H˜共z,z

⬘

兲兴␳共z,z

⬘

^,t兲⫽⫺iបA˜共z⫺z

⬘

兲␳共z,z

⬘

^,t兲. 共31兲 The matrix A˜ is defined as

A˜共␨⫺兲⫽

冉

^{⫺AA共00共0␨兲⫺兲 ⫺AA共00共0␨兲⫺兲 ⫺AA共00共0␨兲⫺兲 ⫺AA共00共0␨兲⫺兲}

冊

_共32^, _兲

where A(␨_⫺⁾⫽(1/ប²)A₀J(␨_⫺^{) and} ␨_⫺⫽z⫺z

⬘

. The equa- tions for the diagonal elements, i.e., ␳00, ␳11, and for the off-diagonal elements, i.e.,␳01, ␳10decouple. However, the decoupling is a consequence of the restriction to the two lowest-energy levels and is not found in the general case.

Yet, the decoupling allows us to find an explicit solution for the time evolution of the diagonal elements. The matrix equation proves to be diagonal for the linear combinations

␳^⫾⬅␳00⫾␳11 and introducing the new coordinates ␨_⫺⫽z

⫺z

⬘

^, ␨_⫹⫽¹2(z⫹z

⬘

) we obtain the general solution

␳k⫾共␨⫺,␨⫹,t兲⫽R_k^⫾

冉

^{␨⫺⫺បmkt}

冊

^{eik␨⫹e⫺⌫k⫾(␨⫺,t)t, 共33兲}

where the decay is described by

⌫k⫾共␨⫺,t兲⫽A共0兲⫿1 t

冕

0

t

dt

⬘

^A

冉

^{␨⫺⫺បmkt}

^⬘ 冊

^{. 共34兲}

The function R_k^⫾is fixed by the initial conditions, i.e., by the density matrix at time t⫽0:

R_k^⫾共␨_⫺兲⫽

冕

⫺⬁

⬁

d␨_⫹^e⫺ik␨⫹␳^⫾共␨_⫺^,␨_⫹^,t⫽0兲. 共35兲 Note that ⌫ in Eq.共34兲 is a function of the spatial variable

␨_⫺ and the wave vector k, and is in general not linear in the time argument t. Adding and subtracting the contributions

␳^⫾finally leads to the following expressions for the diagonal elements of the density matrix:

␳00共␨_⫺^,␨_⫹^,t兲⫽1 2

冕

⫺⬁

⬁ dk

2␲^关␳k⫹共␨_⫺^,␨_⫹^,t兲⫹␳k⫺共␨_⫺^,␨_⫹^,t兲兴. 共36兲 The result for␳11can be obtained from Eq.共36兲by replacing the plus sign between the terms by a minus sign. To get a feeling for the spatial correlations we trace out the center-of- mass coordinate ␨_⫹^:

␳

¯00共␨_⫺^,t兲⫽

冕

⫺⬁

⬁

d␨_⫹␳00共␨_⫺^,␨_⫹^,t兲

⫽e^{⫺[A(0)⫺A(␨⫺)]t1}₂关^¯␳00共␨_⫺^,0兲共1⫹e^{⫺2A(␨⫺)t}兲

⫹^¯␳11共␨_⫺^,0兲共1⫺e^{⫺2A(␨⫺)t}兲兴. 共37兲 Equation 共37兲 shows that we can distinguish two decay mechanisms. There is an overall decay of the spatial off- diagonal elements with a rate ⌫¯

dec(␨_⫺⁾⫽关A(0)⫺A(␨_⫺⁾兴. This decay only affects the density matrix for ␨_⫺⫽0, thus suppressing the spatial coherence. The spatial correlation of the potential fluctuations can be extracted from Fig. 3, show- ing ⌫¯

dec(␨_⫺). We find for the potential fluctuations a corre- lation length ␰cof the order of the trap-to-chip surface dis- tance␰c⬇r₀. The correlation length␰cmust not be confused with the coherence length describing the distance over which the transport of an atom along the trapping well is coherent.

This coherence length is described by the decoherence time and the speed of the moving wave packet. The correlation length ␰c characterizes the distance over which the potential fluctuations are correlated.

The second mechanism describes the equilibration of ex- cited and ground state, quantified by the diagonal elements

␳

¯₀₀(0,t), which occurs at a rate ⌫¯_pop(0)⫽2A(0). Here equilibration means that, due to this mechanism, the prob- ability to be in the ground state, i.e., ␳00(0,t), tends to 1/2.

Of course, at the same time the probability␳11(0,t) to be in the excited state approaches 1/2 at the same rate. Note that for the spatial off-diagonal matrix elements the equilibration rate depends on ␨⫺.

We will now take a closer look onto the quantity A(0) for realistic trap parameters. Using J(0)⫽3␲^/8y0

5and Eq.共26兲, Eq. 共29兲leads to

A共0兲⫽3␲

2 k_BT_eff␴^{A Bbias}

(x)

2ប

冑

^m

冉

^4␮␲0

冊

²

冉

^{2B␮bias(z)BgF}

冊

^3/2r104.

共38兲 The inset of Fig. 3 plots A(0) over r₀ for reasonable trap parameters. Equation共38兲shows that the decay rate is scal- ing on the wire-to-trap distance as 1/r₀⁴ giving a rapid in- crease of decoherence effects once the atomic cloud is brought close to the wire.

As a specific example we want to study the time evolution of the full density matrix in the ground state. Figures 4 and 5 show the time evolution of the absolute value of the density- matrix element兩␳00(␨_⫺^,␨_⫹^,t)兩 for a Gaussian wave packet of spatial extent w_z⫽20r₀ and a wire-to-trap distance of r₀

⫽5 ␮m. Initially, at time t⫽0 all other elements of the den- sity matrix are zero. The time evolution is calculated for the reduced subspace using Eqs. 共33兲 and共36兲. For the spatial correlation we use the approximation共dashed line in Fig. 3兲

A共␨_⫺兲⬇4 A₀

ប²r₀⁵

冋冉

^332␲

冊

^2/3⫹

冉

^␨r⫺0

冊

²

册

^{⫺3/2. 共39兲}

The time evolution of the Gaussian wave packet shows a strip of 兩␨_⫺兩⬍l_c⬇r₀ in which the density matrix decays much slower. This is a consequence of the␨⫺dependence of the decay rate shown in Fig. 3. The spatial correlation length of the wave packet can hence be extracted from Fig. 5 as l_c⬇r₀. In addition, a damped oscillation in the relative co- ordinate␨_⫺ is arising which is getting more pronounced for larger values of ␨⫹. Figure 5 shows cuts along the

␨_⫺-direction for different values of ␨_⫹. The origin of the damped oscillations is the k dependence of the equilibration and decoherence mechanism described by⌫k⫾in Eq.共34兲. To demonstrate the influence of the k-dependent damping we assume that ⌫k⫾⫽A(0)⫿关A(␨_⫺⁾⫹␤⁽␨_⫺^,t)k兴. Choosing ⌫ FIG. 3. Spatial dependence of the decay rate ⌫¯_dec(␨⫺)⫽A(0)

⫺A(␨⫺) for the diagonal elements¯␳_ii(␨⫺,t), Eq.共37兲. The wire- to-trap distance is r₀⫽100␮m and the trap frequency is␻⬇2␲

⫻10 kHz. The dashed line is the approximation, Eq. 共39兲, for A(␨⫺). Inset: Decay rate A(0) as a function of r₀. The parameters chosen in the plots correspond to ⁸⁷Rb trapped at Teff⫽300 K in a magnetic trap with a gold wire of conductivity ␴Au⫽4.54

⫻10⁷⍀^⫺1m^⫺1 and a cross section A⫽2.5⫻5 ␮m². The bias fields chosen are B_bias^(x)⫽80 G and B_bias^(z)⫽2 G.

FIG. 4. Time evolution of兩␳00(␨⫺,␨⫹,t)兩 after t⫽2/A(0). At t⫽0 the wave packet is a Gaussian wave packet of spatial extent w_z⫽20r₀. The wire-to-trap distance is r₀⫽5␮m and all other pa- rameters correspond to those used in Fig. 3. A(0)⬇0.5 s^⫺1is given by Eq.共38兲. The density matrix shows damped oscillations which become more pronounced as␨⫹increases.

FIG. 5. Decay of the absolute value兩␳00(␨⫺,␨⫹⫽30r0,t)兩and 兩␳00(␨⫺,␨⫹⫽0,t)兩 共inset兲. The wave packet is a Gaussian wave packet of spatial extent wz⫽20r0at t⫽0. The wire-to-trap distance is r₀⫽5␮m. All other parameters are the same as used in Fig. 3.

A(0)⬇0.5 s^⫺1 is given by Eq.共38兲. The dashed line关t⫽2/A(0)兴 corresponds to cuts of Fig. 4 along ␨⫺ for ␨⫹⫽30r₀ 共inset ␨⫹

⫽0). Spatial correlations of the potential fluctuations are restricted to a narrow band around ␨⫺⫽0. The width of this band is of the order of r₀. The density matrix shows damped oscillations which become more pronounced with increasing ␨⫹. The increase of 兩␳00(␨⫺,␨⫹⫽30r₀,t)兩for increasing time t is a consequence of the spreading of the wave packet.

linear in k leads to a modulation of the density matrix by a factor proportional to cosh(i␣0␤⫹␣1), describing damped os- cillations.␣0/1are real functions of␨_⫾and t. The oscillations arise only for nonvanishing␤. Decay rates which are k in- dependent do not show oscillations.

The decay described by ⌫k⫾ in Eq. 共34兲 is however not linear, but includes higher powers in k. Hence the simple linear model⌫k⫾⫽A(0)⫿关A(␨⫺)⫹␤⁽␨⫺,t)k兴describes the oscillations in Figs. 4 and 5 only qualitatively.

B. The double-wire trap

The second configuration which we discuss is a double- wire trap关20兴. A system of two parallel trapping wells is of special interest for high-precision interferometry 关20–22兴or beam-splitter geometries 关40兴. The effect of decoherence is one of the key issues in these experiments.

Form and characteristic of the double-wire trap is de- scribed in Refs.关20,32兴so that we will only briefly introduce the main features of the double-wire trapping potential. Fig- ure 1共b兲 shows the setup for the trapping field, which is generated by two infinite wires running parallel to the zˆ axis, separated by a distance d, and a superposed homogeneous bias field B_bias^(x) perpendicular to the current direction. A sec- ond bias field pointing along the zˆ direction, B_bias^(z) , is added in experimental setups to avoid spin flips. Thus the trapping field is

B共x兲⫽␮0I

2␲

冋 ^冉

^x⫹d21

^冊

^2⫹y2

冉

^x⫺⫹0yd2

冊

⫹ 1

冉

^x⫺d2

冊

^2⫹y2

冉

^x⫺⫺0yd2

冊 册

^⫹

^冉

^{BB0bias(x)bias(z)}

^冊

^{. 共40兲}

There are two different regimes for the positions of the trap minima. Defining a critical wire separation ¯y₀

⫽␮0I/(2␲^Bbias

(x)) we can distinguish the situation of having d⬎2 y¯

0 where the trap minima are located on a horizontal line at x₀^L/R⫽⫿

冑

^d2/4⫺¯y

0

2 and y₀⫽¯y₀, and the situation of d⬍2 y¯₀ where the trap minima are positioned on a vertical line at x₀⫽0 and y₀^⫾⫽¯y₀⫾

冑

^¯y0

2⫺d²/4. The two trap minima overlap and form a single minimum for d⫽2 y¯₀.

Further on we will restrict ourselves to the regime d

⬎2 y¯

0 since in this configuration we have two horizontally spaced, but otherwise identical trap minima. The two trap minima form a pair of parallel waveguides. This kind of geometry has been suggested for an interference device 关20,21兴. Figure 6 shows a contour plot of the double trap in the d⬎2y¯₀ regime.

We now analyze the double-wire setup along the lines of Sec. III A. First, an explicit expression for the correlation function 具␦^Sim(z)␦^Sn j(z

⬘

⁾典 is needed. After some calcula- tion we obtain

具␦^Sim共z兲␦^Snj共z

⬘

兲典

⫽A₀ 8 w²

x₀^␣x₀^␤

d² _␥⫽

兺

_L,R^{J␣␤␥ 共z⫺z}

^⬘

^兲

冕

^dr⬜

冕

^dr⬜

^⬘

⫻␹i共r_⬜兲

冋

^{共y⫺y0兲⫺}

^冉

^{yx00␣共⫹x⫺⑀x␥0␣d2兲}

^冊 册

^{␹m*共r⬜兲}

⫻␹n共r_⬜

⬘

兲

冋

^共y

^⬘

^⫺y0兲⫺

^冉

^{yx0共0␤x⫹}

^⬘

^{⫺⑀␥xd20␤}

^冊

^兲

册

^␹j*共r⬜

^⬘

^{兲, 共41兲}

where the spatial correlation is now given by

J_␣␤^␥ 共z⫺z

⬘

兲⫽

冕

⫺⬁

⬁

dz˜

冋冉

^{x0␣⫺⑀␥d2}

冊

^{2⫹y02⫹˜z2}

册

^⫺3/2

⫻

冋冉

^{x0␤⫺⑀␥d2}

冊

^{2⫹y02⫹共z⫺z}

^⬘

^⫺˜z兲2

册

^⫺3/2.

共42兲 Here,␣is the trap label corresponding to i and m, and␤^the trap label corresponding to n and j, the indices which occur in 具␦^Sim(z)␦^Sn j(z

⬘

⁾典. The wires are located at x⫽d_L/R⫽

⫿d/2 and ⑀_␥ is defined as⑀L⫽⫺1, ⑀R⫽1. The result, Eq.

共41兲, uses the assumption that the extent of the wave function w is much smaller than y₀ and much smaller than the trap FIG. 6. Contour plot of the magnetic field in the double-wire geometry for B_bias^(x)⫽10 G, B_bias^(z)⫽5 G, and I⫽0.3 A. The wires are located at x⫽⫾100␮m and y⫽0. Two potential minima are lo- cated roughly above the current-carrying wires. The upper 共right兲 plot shows a horizontal 共vertical兲 cut through the right potential minimum. The dashed lines in the upper and the right plot show the harmonic approximation to the trapping field.